Homogeneous System: Why Invertible A Has No Non-Zero Solutions?

In summary, the conversation discusses the conditions for a homogeneous system to have a non-trivial solution. It is stated that for this to occur, the coefficient matrix must be singular. It is then questioned why an invertible matrix would not have any non-zero solutions. The response explains that if A is invertible, then A-1 exists and the system will only have the trivial solution x=0. This is because both A and A-1 must be one-to-one, preventing any additional solutions.
  • #1
cupu
6
0
Hello,

In a book I'm reading about linear algebra it's mentioned that in order for the homogeneous system
Ax = 0
to have a solution (other than the trivial solution) the coefficient Matrix must be singular.
The thing is, I can't remember (the wikipedia page on homogeneous systems didn't turn up anything) why if A is invertible, then the system does not have non-zero solutions.

Any help on why this is so is appreciated.

Thank you in advance
 
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  • #2
If A is invertible, then A-1 exists.
Ax = 0 ==> A-1Ax = A-10 = 0 ==> x = 0

Another way to look at it is that, for A-1 to exist, both it and A must be one-to-one. The equation Ax = 0 obviously has at least one solution, x = 0, but the one-to-oneness prevents it from having any additional solutions.
 
  • #3
That makes sense, thank you very much for the response Mark44.
 

1. What is a homogeneous system?

A homogeneous system is a system of linear equations where the constant terms are all equal to zero. In other words, the right-hand side of each equation is zero.

2. Why is it important to study homogeneous systems?

Homogeneous systems have many applications in fields such as physics, engineering, and economics. They also serve as a foundation for understanding more complex systems and solving real-world problems.

3. What does it mean for a matrix to be invertible?

A matrix is invertible if it has an inverse matrix, which is a matrix that, when multiplied by the original matrix, results in the identity matrix. In other words, the inverse undoes the original matrix.

4. Why does an invertible matrix have no non-zero solutions?

An invertible matrix has no non-zero solutions because its inverse can only multiply with the original matrix to produce the identity matrix. Therefore, the only solution to the homogeneous system is the trivial solution, where all variables are equal to zero.

5. How can I determine if a matrix is invertible?

A square matrix is invertible if its determinant is non-zero. This means that the matrix has a unique solution or no solution at all. Additionally, a matrix can be tested for invertibility through row reduction or by checking if all its columns are linearly independent.

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